A116532 -id:A116532 - cp's OEIS frontend

A089674 a(n) = number of n X n (0,1) matrices A such that the 2n+2 vectors consisting of the rows and the columns of the matrix A, as well as the main diagonal read in the upward direction and the main antidiagonal, are all distinct.

0, 0, 0, 1692, 2329280, 13441654352, 190945826194432

Offset: 1

Vladeta Jovovic, Jan 04 2004

Binary matrices with distinct rows and columns, various versions: A059202, A088309, A088310, A088616, A089673, A089674, A093466, A094000, A094223, A116532, A116539, A181230, A259763

a(6)-a(7) from Bert Dobbelaere, May 05 2025

A094223 Number of binary n X n matrices with all rows (columns) distinct, up to permutation of columns (rows).

Original entry on oeis.org

1, 2, 7, 68, 2251, 247016, 89254228, 108168781424, 451141297789858, 6625037125817801312, 348562672319990399962384, 66545827618461283102105245248, 46543235997095840080425299916917968, 120155975713532210671953821005746669185792, 1152009540439950050422144845158703009569109376384

Offset: 0

Goran Kilibarda and Vladeta Jovovic, May 28 2004

Andrew Howroyd, Table of n, a(n) for n = 0..50
Goran Kilibarda and Vladeta Jovovic, Enumeration of some classes of T_0-hypergraphs, arXiv:1411.4187 [math.CO], 2014.

Main diagonal of A059584 and A059587, A060690, A088309.

Binary matrices with distinct rows and columns, various versions: A059202, A088309, A088310, A088616, A089673, A089674, A093466, A094000, A094223, A116532, A116539, A181230, A259763

a[n_] := Sum[(-1)^(n - k)*StirlingS1[n, k]*Binomial[2^k, n], {k, 0, n}]; (* or *) a[n_] := Sum[ StirlingS1[n, k]*Binomial[2^k + n - 1, n], {k, 0, n}]; Table[ a[n], {n, 0, 12}] (* Robert G. Wilson v, May 29 2004 *)

a(n) = sum(k=0, n, stirling(n, k, 1)*binomial(2^k+n-1, n)); \\ Michel Marcus, Dec 17 2022

a(n) = Sum_{k=0..n} (-1)^(n-k)*Stirling1(n, k)*binomial(2^k, n).

a(n) = Sum_{k=0..n} Stirling1(n, k)*binomial(2^k+n-1, n).

More terms from Robert G. Wilson v, May 29 2004

a(13) onwards from Andrew Howroyd, Jan 20 2024

A000410 Number of singular n X n rational (0,1)-matrices.

Original entry on oeis.org

0, 0, 6, 425, 65625, 27894671, 35716401889, 144866174953833

Offset: 1

N. J. A. Sloane

Number of all n X n (0,1)-matrices having distinct, nonzero ordered rows and determinant 0 - compare A000409.

a(n) is the number of singular n X n rational {0,1}-matrices with no zero rows and with all rows distinct, up to permutation of rows and so a(n) = binomial(2^n-1,n) - A088389(n). Cf. A116506, A116507, A116527, A116532. - Vladeta Jovovic, Apr 03 2006

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. Metropolis and P. R. Stein, On a class of (0,1) matrices with vanishing determinants, J. Combin. Theory, 3 (1967), 191-198.
Miodrag Zivkovic, Classification of small (0,1) matrices, arXiv:math/0511636 [math.CO], 2005.
Miodrag Zivkovic, Classification of small (0,1) matrices, Linear Algebra and its Applications, 414 (2006), 310-346.
Index entries for sequences related to binary matrices

Cf. A000409, A046747, A064230, A064231.

n! * a(n) = A046747(n) - 2^(n^2) + n! * binomial(2^n -1, n).

n=7 term from Guenter M. Ziegler (ziegler(AT)math.TU-Berlin.DE)

a(8) from Vladeta Jovovic, Mar 28 2006

A116506 Number of singular n X n rational {0,1}-matrices with no zero rows.

Original entry on oeis.org

0, 3, 169, 28065, 16114831, 33686890209, 262530190180063, 7717643584470877185

Offset: 1

Vladeta Jovovic, Apr 03 2006

Cf. A000410, A116507, A116527, A116532.

a(n) = A055601(n) - A055165(n).

A116527 Number of singular n X n rational {0,1}-matrices with no zero rows or columns and with all rows distinct and all columns distinct, up to permutation of rows.

Original entry on oeis.org

0, 0, 0, 75, 22365, 13303500, 21058940420, 98692672142610

Offset: 1

Vladeta Jovovic, Apr 03 2006

Cf. A000409, A000410, A116506, A116507, A116532.

a(n) = A094000(n) - A088389(n).

Conjecture: a(n) = A000410(n) - A000409(n-1) for n>1. - Jean-François Alcover, Jan 08 2020

A116507 Number of singular n X n rational {0,1}-matrices with no zero rows or columns.

Original entry on oeis.org

0, 1, 91, 18943, 12483601, 28530385447, 235529139302185, 7183142489571818623

Offset: 1

Vladeta Jovovic, Apr 03 2006

Cf. A000410, A116506, A116527, A116532.

a(n) = A048291(n) - A055165(n).

cp's OEIS Frontend

A089674 a(n) = number of n X n (0,1) matrices A such that the 2n+2 vectors consisting of the rows and the columns of the matrix A, as well as the main diagonal read in the upward direction and the main antidiagonal, are all distinct.

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A094223 Number of binary n X n matrices with all rows (columns) distinct, up to permutation of columns (rows).

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A000410 Number of singular n X n rational (0,1)-matrices.

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A116506 Number of singular n X n rational {0,1}-matrices with no zero rows.

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A116527 Number of singular n X n rational {0,1}-matrices with no zero rows or columns and with all rows distinct and all columns distinct, up to permutation of rows.

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A116507 Number of singular n X n rational {0,1}-matrices with no zero rows or columns.

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